# Curry Triangle

• Aug 25th 2010, 01:34 AM
tsang
Curry Triangle
Hi, can anyone help me with this please? I want to come out with a formal proof that Curry Triangle paradox. I know the truth is that actually the side is not connected, there is slightly change on the side length. But, how can I come out with a reasonable proof? Thanks a lot.
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• Aug 25th 2010, 08:25 AM
Sudharaka
Quote:

Originally Posted by tsang
Hi, can anyone help me with this please? I want to come out with a formal proof that Curry Triangle paradox. I know the truth is that actually the side is not connected, there is slightly change on the side length. But, how can I come out with a reasonable proof? Thanks a lot.
I have attached the shape with the message.

Dear tsang,

How about the solution that is stated in wikipedia?(Missing square puzzle - Wikipedia, the free encyclopedia) I mean you can show that the ratio of sides in the blue and red traingles are not same. Hence the whole figure is an optical illusion.
• Aug 25th 2010, 09:17 AM
Soroban
Hello, tsang!

The Curry Triangle paradox is based on two non-similar triangles.

The two right triangles appear to be similar (and interchangeable),
. . but they are not.

If they were similar right triangles, they would have a common hypotenuse.
(Designated o - o - o - o ...)

Code:

```                                      o                                 ..o.* |                             ..o:::*  |3                         ..o:::::*    |                     ..o:::::::* - - - *                 ..o:::::*    |  5             .o:::*          |3           o:*                |       * - - - - - - - - - - - *                   8```

In the first diagram, the two triangles fall below the common hypotenuse.
Their area is short by exactly one-half a square unit.

Code:

```                                  ...o                             ...*:o  |                       ...*:::o      |3                 ...*:::::o          |               * - - - o - - - - - - - *           .*:|  o            8         .*:::o2       .*:o  |       o - - - *           5```

In the second diagram, the triangle are partly above the common hypotenuse.
Their area is over by exactly one-half a square unit.

And that is the basis of the paradox.